Worksheet: Graphs of Exponential Functions

In this worksheet, we will practice sketching and identifying the graphical transformations of exponential functions.

Q1:

Which graph demonstrates exponential growth?

A

B

C

D

Q2:

Determine the point at which the graph of the function 𝑓(𝑥)=6 intersects the 𝑦-axis.

A(6,0)

B(1,0)

C(0,1)

D(0,6)

Q3:

Determine the function represented by the graph shown.

A𝑦=−2

B𝑦=2

C𝑦=2

D𝑦=−2

Q4:

The graph of the function 𝑓(𝑥)=𝑎 passes through the point
(3,1). What is the value of 𝑎?

Q5:

Which of the following expressions does NOT describe the shown graph?

A𝑁=200⋅𝑒

B𝑁=200⋅(4)

C𝑁=200⋅12

D𝑁=200⋅14

E𝑁=200⋅12

Q6:

Which of the following expressions does NOT describe the shown graph?

A𝑁=120⋅12lnln

B𝑁=120⋅(3)

C𝑁=120⋅12lnln

D𝑁=120⋅13

E𝑁=120⋅𝑒ln⋅

Q7:

Which of the following graphs represents the equation 𝑦=−4(2)?

A

B

C

D

Q8:

Which of the following graphs represents the equation 𝑦=14?

A

B

C

D

E

Q9:

Which of the following could be the equation of the curve?

A𝑦=−4(1+3)

B𝑦=4(1+3)

C𝑦=4(1−3)

D𝑦=4(1+3)

E𝑦=4(1−3)

Q10:

Which of the following graphs represents the equation
𝑦=23?

A

B

C

D

E

Q11:

Which of the following graphs represents the equation 𝑦=2(3)?

A

B

C

D

E

Q12:

Which of the following graphs represents the equation
𝑦=5332?

A

B

C

D

E

Q13:

Which of the following graphs represents the equation 𝑦=23(3)?

A

B

C

D

E

Q14:

Among these expressions, which one does NOT describe the shown graph?

A𝐴=50⋅4

B𝐴=50⋅6

C𝐴=50⋅2

D𝐴=50⋅8

E𝐴=50⋅10

Q15:

Which of the following graphs represents the equation 𝑦=3?

A

B

C

D

E

Q16:

Which of the graphs is that of 𝑓(𝑥)=2⋅3?

AD

BA

CB

DC

Q17:

Complete the sentence:
The graph of an exponential function 𝑓(𝑥)=𝑎 with 𝑎>0
and 𝑎≠1.

Acontains the point (0, 1)

Bhas a horizontal asymptote at 𝑦=1

Cusually has both positive and negative 𝑦-values

Dhas a domain of positive real numbers

Q18:

Which of the graphs is that of 𝑓(𝑥)=4⋅(12)?

AB

BC

CA

DD

Q19:

To get the graph of 𝑦=5 from the graph of 𝑦=2, we must .

Ascale by a factor of loglog(5)(2) in the horizontal direction.

Bscale by a factor of 52 in the horizontal direction.

Cscale by a factor of loglog(2)(5) in the vertical direction.

Dscale by a factor of loglog(2)(5) in the horizontal direction.

Escale by a factor of loglog(5)(2) in the vertical direction.

Q20:

Observe the given graph, and then answer the following questions.

Find the 𝑦-intercept in the shown graph.

As this graph represents an exponential function, every 𝑦-value is multiplied by 𝑏 when 𝑥 increases by Δ𝑥. Find 𝑏 for Δ𝑥=2.

Find the equation that describes the graph in the form 𝑦=𝑎𝑏.

A𝑦=0.5⋅3

B𝑦=0.5⋅3

C𝑦=0.5⋅3

D𝑦=0.5⋅2

E𝑦=0.5⋅3

Q21:

Which of the following graphs represents the equation 𝑦=−34(2)?

A

B

C

D

Q22:

Mason argues that just as two data points are enough to uniquely determine a linear function, so too is an exponential function uniquely determined by two points on its graph. Is this true?

Ayes

Bno

Q23:

A close-up of the graphs of 𝐸(𝑥)=𝑒
(dashed) against 𝑃(𝑥)=𝑥 (solid) below shows that even though
𝑒=1>0=0, this quickly reverses.

Indeed, at 𝑥=2, the power
function is much larger, where 𝑃(2)=1,024
while 𝐸(2)≈7.39 and 𝑃(5)≈10 while
𝐸(5)≈148.

Which function is larger at 𝑥=10,
the power function or the exponential function?

AExponential

BPower

The curves below show the graph of ln𝑥 which is
the same as 10𝑥ln
above that of ln𝑒
which simplifies to 𝑥.

What do the indicated points tell you about 𝑃(11)
and 𝐸(11)?

A𝑃(11)=𝐸(11)

B𝑃(11)<𝐸(11)

C𝑃(11)>𝐸(11)

A fact about the natural logarithm function is that the slope of its tangent at
(𝑥,𝑥)ln is just 1𝑥.
For the
constant multiple 𝐿(𝑥)=10𝑥ln,
the tangent has slope
10𝑥 at (𝑥,10𝑥)ln.
What is the
slope of this tangent line at the point (20,𝐿(20))?
What is the
equation of the line?

ASlope =12, line: 𝑦=12𝑥−10+(20)ln

BSlope =2, line: 𝑦=2𝑥−10+10(20)ln

CSlope =12, line: 𝑦=12𝑥−20+10(20)ln

DSlope =2, line: 𝑦=2𝑥−20+10(20)ln

ESlope =12, line: 𝑦=12𝑥−10+10(20)ln

Since that tangent had a slope less than 1, it is bound to meet 𝑦=𝑥 at some point. What is the 𝑥-coordinate of this
point to the nearest integer?

A10(20)−10≈20ln

B10(10)−10≈13ln

C20(20)+20≈80ln

D10(10)+10≈33ln

E20(20)−20≈40ln

By considering the convexity of 𝐿(𝑥),
what can we conclude about 𝑃(40) and 𝐸(40).

A𝐸(40)<𝑃(40)

B𝐸(40)>𝑃(40)

C𝐸(40)>𝑃(40)

D𝐸(40)=𝑃(40)

E𝐸(40)<𝑃(40)

Use this method of finding the equation of
a tangent with a slope of 12, to find an integer 𝑛 such
that 𝑒>𝑛.

A𝑛=480

B𝑛=632

C𝑛=350

D𝑛=362

E𝑛=463

Use this method, but this time with a slope 14, to find an
integer 𝑛 such that 𝑒>𝑛.

A𝑛=463

B𝑛=350

C𝑛=493

D𝑛=362

E𝑛=480

Q24:

Assuming 𝑎>1, which axis is the graph of the exponential
function 𝑓(𝑥)=𝑎 asymptotic to?

AThe positive 𝑦-axis

BThe negative 𝑦-axis

CThe positive 𝑥-axis

DThe negative 𝑥-axis

Q25:

If 0<𝑎<1, for which values of
𝑥 does the exponential function 𝑎 satisfy
0<𝑎<1?